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What you want is known as structural induction, or more vaguely as "the induction principle implied by the inductive definition".

The crucial point is that every element of $\Lambda$ is there by virtue of one of your three rules -- this is sometimes hinted at by a fourth rule reading: "Nothing else is in $\Lambda$". Your inductive definition is really shorthand for the following set-theoretic construction:

Let $\Lambda$ be the union of all $A_n$. In other words $\Lambda=\bigcup_{n\in \mathbb N}A_n$.

Now, the structural induction principle says: In order to prove that some property holds for all $x\in\Lambda$ it suffices to prove for an arbitrary $A$ that if all $x\in A$ has the property, then the property holds for all $a\in\Phi(A)$.

This can principle can be justified by ordinary mathematical induction on $n$ for the statement that the property holds for every $x\in A_n$. When this is true for all $n$, then since every $x\in\Lambda$ is in some $A_n$, the property holds for every such $x$.

@kjo: To put it a little more colorfully, you can think of the $n$ at which a member of $\Lambda$ first appears in Henning’s construction as its ‘birthday’. The induction is then on the birthdays of the elements of $\Lambda$: if $\ell(x)>0$ for every $x$ whose birthday precedes $n$, then the construction of $\Lambda$ ensures that $\ell(x)>0$ for every $x$ whose birthday is $n$.
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Brian M. ScottDec 25 '11 at 1:59

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@Henning: Perhaps you should also say something about why we don't need the full power of transfinite/well-founded induction here. (It has to do with the fact that all the constructors are finitary.)
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Zhen LinDec 25 '11 at 4:34